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Hawkes Learning Systems:
College Algebra
Section 2.4: Higher Degree Polynomial
Equations
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Objectives
o Solving quadratic-like equations.
o Solving general polynomial equations by factoring.
o Solving polynomial-like equations by factoring.
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Solving Quadratic-Like Equations
o A polynomial equation of degree n in one variable,
say x , is an equation that can be written in the form
a n x  a n 1 x
n
n 1
 ...  a1 x  a 0  0
where a i is a constant and a n  0 .
o In general, there is no method for solving polynomial
equations that is guaranteed to find all solutions.
o Since the Zero-Factor property applies whenever a
product of any finite number of factors is equal to 0,
we can use this property to solve quadratic-like
equations.
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Solving Quadratic-Like Equations
An equation is quadratic-like, or quadratic in
form, if it can be written in the form
aA  bA  c  0
2
Where a , b , and c are constants, a  0 , and A is
an algebraic expression. Such equations can be
solved by first solving for A and then solving for
the variable in the expression A .
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Example 1: Solving Quadratic-Like Equations
Solve the quadratic-like equation.
x
Step 1: Let
 3x
2

2

2
2
 A  4 A  2  0
and factor.
x  3x
2
and solve
for x .
2
A  2A 8  0
A  x  3x
Step 2: Replace
A with

 2 x  3x  8  0
A4
or
A  2
x  3x  4
or
x  3 x  2
x  3x  4  0
or
x  3x  2  0
 x  4   x  1  0
or
 x  2   x  1  0
x   4 or x  1
or
2
2
2
2
x  2 or x  1
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Example 2: Solving Quadratic-Like Equations
Solve the quadratic-like equation.
2
1
x 3  5x3  6  0
2
 3
 13 
 x   5 x   6  0




1
 1 
 1 

3
3
  x   1    x   6   0





1
1
x 3  1
x  (  1)
x  1
3
or
x3  6
or
or
x6
3
x  216
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Solving General Polynomial Equations by
Factoring
o If an equation of the following type can be factored
completely, then the equation can be solved by using
the Zero-Factor Property.
a n x  a n 1 x
n
n 1
 ...  a1 x  a 0  0
o If the coefficients in the polynomial are all real, the
polynomial can, in principle, be factored.
o In practice, this may be difficult to accomplish unless
the degree of the polynomial is small or the
polynomial is easily recognizable as a special
product.
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Example 3: General Polynomial Equations
Solve the equation by factoring.
4
x 40
4
Step 1: Isolate
0 on one
side and
factor.
Step 2: Set both
equations
equal to 0
and solve.
x 4
x
x 20
2
x  2
2
x  i 2
2
 2  x  2   0
2
or
or
or
x 20
2
x 2
2
x 2
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Example 4: General Polynomial Equations
Solve the equation by factoring.
z  z  9z  9  0
3
z
2
2
 z  1  9  z  1  0
 z  1  z  9   0
2
 z  1  z  3   z  3   0
z 1 0
or z  3  0
z   1 or
z3
or z  3  0
or
z  3
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Example 5: General Polynomial Equations
Solve the equation by factoring.
64 t  8  0
3
(4 t )  2  0
3
3
 4 t  2  16 t 2  8 t  4   0
4t  2t  1  0
2
4t  2  0
4t  2
t
1
2
or
or
or
t
t
2  4  4  1 
2 
2
24
2 
 12
8
t
1  i 3
4
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Solving Polynomial-Like Equations by Factoring
o Some equations that are not polynomial can be
solved using the methods we have developed in
Section 2.4.
o The general idea will be to rewrite the equation so
that 0 appears on one side, and then to apply the
Zero-Factor Property.
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Example 6: Polynomial-Like Equations
Solve the following equation by factoring.
11
6
1
x 5  3x 5  4x 5
11
Step 1: Isolate 0 on
one side.
6
1
x 5  3x 5  4 x 5  0
1
Step 2: Factor.
x 5  x  3x  4  0
2
1
x 5  x  4   x  1  0
Step 3: Apply the
Zero-Factor
Property.
1
x5  0
x0
x40
x  4
x 1  0
x 1
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Example 7: Polynomial-Like Equations
Solve the following equation by factoring.
 x  3
 x  3

1
2

1
2
1
 2  x  32  0
1  2  x  3    0

1
 x  3 2  2 x  5  0
 x  3

1
2
0
or
2x  5  0
1
0
or
2x  5
1
 x  32
No Solution
x
5
2